Electron. J. Diff. Equ., Vol. 2011 (2011), No. 114, pp. 1-11.

p-harmonious functions with drift on graphs via games

Alexander P. Sviridov

In a connected finite graph $E$ with set of vertices $\mathfrak{X}$, choose a nonempty subset, not equal to the whole set, $Y\subset \mathfrak{X}$, and call it the boundary $Y=\partial\mathfrak{X}$. Given a real-valued function $F: Y\to \mathbb{R}$, our objective is to find a function $u$, such that $u=F$ on $Y$, and for all $x\in \mathfrak{X}\setminus Y$,
 u(x)=\alpha \max_{y \in S(x)}u(y)+\beta \min_{y \in S(x)}u(y)
 +\gamma \Big( \frac{\sum_{y \in S(x)}u(y)}{\#(S(x))}\Big).
Here $\alpha, \beta,  \gamma $ are non-negative constants such that $\alpha+\beta + \gamma =1$, the set $S(x)$ is the collection of vertices connected to $x$ by an edge, and $\#(S(x))$ denotes its cardinality. We prove the existence and uniqueness of a solution of the above Dirichlet problem and study the qualitative properties of the solution.

Submitted October 26, 2010. Published September 6, 2011.
Math Subject Classifications: 35Q91, 35B51, 34A12, 31C20.
Key Words: Dirichlet problem; comparison principle; mean-value property; stochastic games; unique continuation.

Show me the PDF file (251 KB), TEX file, and other files for this article.

Alexander P. Sviridov
Department of Mathematics, University of Pittsburgh
Pittsburgh, PA 15260, USA

Return to the EJDE web page